Question

The number of values of \(z \in \mathbb{C}\), satisfying the equations \(|z - (4 + 8i)| = \sqrt{10}\) and \(|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}\), is:

• A. 0
• B. 2
• C. 1
• D. 4

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The problem requires finding the number of intersection points between two loci in the complex plane. The first equation represents a circle, and the second represents an ellipse.
Step 2: Key Formula or Approach:
Identify the geometric parameters of each locus:
- Circle: \( |z - z_0| = r \) has center \( z_0 \) and radius \( r \).
- Ellipse: \( |z - z_1| + |z - z_2| = 2a \) has foci \( z_1, z_2 \) and major axis \( 2a \).
The center of the ellipse is the midpoint of its foci. Calculate the semi-minor axis \( b \) using \( b^2 = a^2 - c^2 \), where \( 2c \) is the distance between the foci.
Step 3: Detailed Explanation:
The first equation is \( |z - (4 + 8i)| = \sqrt{10} \).
This is a circle with center \( C_1(4, 8) \) and radius \( R = \sqrt{10} \).
The second equation is \( |z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5} \).
This represents an ellipse with foci \( F_1(3, 5) \) and \( F_2(5, 11) \).
Let's find the distance between the foci:
\( 2c = |F_1 - F_2| = |(3 - 5) + i(5 - 11)| = |-2 - 6i| = \sqrt{(-2)^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \).
So, the semi-focal distance is \( c = \sqrt{10} \).
The major axis is \( 2a = 4\sqrt{5} \implies a = 2\sqrt{5} \implies a^2 = 20 \).
The semi-minor axis \( b \) is given by \( b^2 = a^2 - c^2 = 20 - (\sqrt{10})^2 = 20 - 10 = 10 \).
Thus, \( b = \sqrt{10} \).
Now, let's find the center of the ellipse:
Center \( C_2 \) is the midpoint of \( F_1 \) and \( F_2 \): \( C_2 = \left(\frac{3+5}{2}, \frac{5+11}{2}\right) = (4, 8) \).
Observe that the circle and the ellipse share the exact same center \( (4, 8) \).
Moreover, the radius of the circle \( R = \sqrt{10} \) is exactly equal to the semi-minor axis \( b = \sqrt{10} \) of the ellipse.
Because they share the same center and the circle's radius equals the ellipse's semi-minor axis, the circle is inscribed inside the ellipse, touching it exactly at the two extremities of the minor axis.
Therefore, they intersect at exactly 2 points.
Step 4: Final Answer:
The number of values of \( z \) is 2.